Gauge transformation symmetry

The development just given illustrates the fact that the topology of the vacuum determines the nature of the gauge transformation, field tensor, and field equations, as inferred in Section (I). The covariant derivative plays a central role in each case for example, the homogeneous field equation of 0(3) electrodynamics is a Jacobi identity made up of covariant derivatives in an internal 0(3) symmetry gauge group. The equivalent of the Jacobi identity in general relativity is the Bianchi identity. 

Noether s theorem for gauge symmetry For a local infinitesimal gauge transformation about a solution of the field equations,... 

Chemical behaviour depends on chemical potential and electromagnetic interaction. Both of these factors depend on the local curvature of space-time, commonly identified with the vacuum. Any chemical or phase transformation is caused by an interaction that changes the symmetry of the gauge field. It is convenient to describe such events in terms of a Lagrangian density which is invariant under gauge transformation and reveals the details of the interaction as a function of the symmetry. The chemically important examples of crystal nucleation and the generation of entropy by time flow will be discussed next. The important conclusion is that in all cases, the gauge field arises from a symmetry of space-time and the nature of chemical matter and interaction reduces to a function of space-time structure. 

The second fundamental symmetry property of the Lagrangian is established by local gauge transformations of the form... 

The gauge transformations form a group. It is Abelian, i.e. diflferent transformations of the group commute with each other, and it is onedimensional, i.e. the transformations are specified by one parameter 0. This group is C/(l), the group of unitary transformations in one dimension. We say that 17(1) is a symmetry group of , and that the functions form a one-dimensional representation of U 1). 

It was demonstrated by Higgs [50] that the appearance of massless bosons can be avoided by combining the spontaneous breakdown of symmetry under a compact Lie group with local gauge symmetry. The potential V() which is invariant under the local transformation of the charged field... 

Non-Abelian electrodynamics has been presented in considerable detail in a nonrelativistic setting. However, all gauge fields exist in spacetime and thus exhibits Poincare transformation. In flat spacetime these transformations are global symmetries that act to transform the electric and magnetic components of a gauge field into each other. The same is the case for non-Abelian electrodynamics. Further, the electromagnetic vector potential is written according to absorption and emission operators that act on element of a Fock space of states. It is then reasonable to require that the theory be treated in a manifestly Lorentz covariant manner. 

This non-Abelian gauge theory satisfies the usual transformation properties. If Jl is the base manifold in four dimensions, then the gauge theory is determined by an internal set of symmetries described by a principal bundle. Let Ua, where a = 1,2,be an atlas of charts on the Ji. The transitions from one chart to another is given by gap f/p —> Ua, where these determine the transition functions between sections on the principal bundle. The transform between one section to another is given by... 

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